Invariance Principles for Ergodic Systems with Slowly $$\alpha $$-Mixing Inducing Base

Chen, Jianyu; Nguyen, Kien Huu

doi:10.1007/978-3-030-01476-6_9

Cited by 2 publications

(1 citation statement)

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“…However, when f is an unbounded observable, the CLT and ASIP may fail for some obvious reasons, for instance, f ∈ L 2 (µ) and thus the corresponding process {f • T n } n≥0 has no finite variance. In order to establish the limiting theorems for such process, we need some to add some extra conditions, such as moment controls in [11,12]. In this paper, we impose the following conditions on the dynamically Hölder series f ∈ H W,γ,t .…”

Section: Stochastic Propertiesmentioning

confidence: 99%

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

Chen¹,

Zhang²,

Zhang³

2020

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In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov's one-step expansion at q-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic Z function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem). For the first time, we are able to make a direct relation between the mixing rates and the Z function, see (2.7). The novelty of our results relies on establishing the regularity of invariant density, as well as verifying the stochastic properties for a large class of unbounded observables.Finally, we verify our assumptions for several well known examples that were previously studied in the literature, and unify results to these examples in our framework.

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Section: Stochastic Propertiesmentioning

confidence: 99%